I am aware of the redox chemistry that happens at the electrode surface, but I'm not sure how exactly a change in electrical potential in the solution would change the potential in the electrode?
It all come down to attractions of electrons to various nuclei. In the case of electrodes there must be intimate mixing of all components so that the transfer of electrons is kinetically facile. It tends to be an art. A potential develops between the cathode, reduction, reaction and the anode, oxidation, reaction and if there is electrical connection current will flow allowing the reactions to occur. The single electrode potential between the Ag and Ag+ ions has been difficult to measure. [So far, I think, impossible.]